If \((A,I)\)  is not a transitive forest, then \(\mathsf {P4}\)  or \(\mathsf {C4}\)  is an induced subgraph of \((A,I)\)  [1]}.
Thus, \(\mathbb {G}(\mathsf {P4})\)  or \(\mathbb {G}(\mathsf {C4}) \cong F_2 \times F_2\)  is a subgroup of \(\mathbb {G}(A,I)\) . Hence,
NP-hardness of knapsack for \(\mathbb {G}(A,I)\)  follows from [2]} or Lemma REF .
